COE 561, Term 101
Digital System Design and Synthesis
HW# 3 Solution
Due date: Tuesday, Dec. 14
Q.1.
Consider the following function:
x = a d e + a f + a' b' c f + b c d e + b c f + c d' e' f
(i) Compute all the kernels of X using the recursive kernel computation
algorithm. Show all the steps.
(ii) Compute all the kernels of X based on matrix representation. Compare your
answer to the result obtained in (i).
(iii) Use the sis command print_kernel and compare the kernels obtained to your
answers in (i) and (ii).
(iv) Find a good factor of X. Assume that input variables are sorted in
lexicographic order. Determine the number of literals obtained. Compare
your solution with the result obtained by running the sis commands factor –g
x; print_factor; print_stats –f.
Q.2.
Consider the following function:
x = a b c d + a b c' d' + a b' e + a b' f + a' b e + a' b f + a' b' c d + a' b' c' d' + c
e' f' + d e' f'
(i) Compute all double-cube divisors of x along with their bases and their
weights. Show only double-cube divisors that have non-empty bases.
(ii) Apply the fast extraction algorithm based on extracting double-cube divisors
along with complements or single-cube divisors with two-literals. Show all
steps of the algorithm. Determine the number of literals saved. Compare your
solution with the result obtained by running the sis commands fx.
Q.3.
Consider the logic network defined by the following expressions:
X = A + C;
Y = A X + X’ B;
Z = Y + X;
Inputs are {A, B, C} and output is {Z}.
(i) Compute the CDC set for the cut at the inputs of circuit Y.
(ii) Compute the ODC set for node Y.
(iii) Simplify the function of Y using both its ODC and CDC.
(iv) Apply the sis command full_simplify and compare the solution obtained with
your obtained solution based in (iii).
Q.4.
Consider the logic network defined by the following expressions:
g=ab
h = a’ b’
i=g+h
j=cd
k=ije
l=ijf
x =k + l
Inputs are {a, b, c, d, e, f} and output is {x}. Assume that the delay of a gate is
related to the number of its inputs. Also, assume that the input data-ready times are
zero except for input a, which is equal to 2.
(i) Draw the logic network graph and compute the data ready times and slacks
for all vertices in the network.
(ii) Determine the maximum propagation delay and the topological critical
path.
(iii) Suggest an implementation of the function x to reduce the delay of the
circuit. What is the maximum propagation delay after the modified
implementation?
(iii) Computing kernels using SIS:
sis> read_eqn hw3q1.eqn
sis> print
{X} = a d e + a f + a' b' c f + b c d e + b c f + c d' e' f
sis> print_kernel
Kernels of {X}
(a) * (d e + f)
(b c) * (d e + f)
(c) * (a' b' f + b d e + b f + d' e' f)
(c f) * (a' b' + b + d' e')
(d e) * (a + b c)
(f) * (a + a' b' c + b c + c d' e')
(-1-) * (a d e + a f + a' b' c f + b c d e + b c f + c d' e' f)
Good Factoring using SIS:
sis> print
{X} = a d e + a f + a' b' c f + b c d e + b c f + c d' e' f
sis> factor -g X
sis> print_factor
{X} = c f (d' e' + a' b') + (b c + a) (f + d e)
sis> print_stats –f
hw3q1.eqn pi= 6 po= 1 node= 1 latch= 0 lits(sop)= 20 lits(ff)= 12
Fast Extraction using SIS:
sis> read_eqn hw3q2.eqn
sis> print
{x} = a b c d + a b c' d' + a b' e + a b' f + a' b e + a' b f + a' b' c
a' b' c' d' + c e' f' + d e' f'
sis> fx
sis> print
{x} = [1] [3] + [1]' [2]' + [1]' c d + [2] [3]'
[1] = a b' + a' b
[2] = c + d
[3] = e + f
sis> print_stats
hw3q2.eqn pi= 6 po= 1 node= 4 latch= 0 lits(sop)= 17 lits(ff)= 16
Simplification of Y using SIS:
sis> read_eqn hw3q3.eqn
sis> print
{Z} = X + Y
X=A+C
Y = A X + B X'
sis> full_simplify
sis> print
{Z} = X + Y
X=A+C
Y=B